A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity

Authors:
Ziran Tu;Yingpu Deng
Affiliations:
Faculty of Science, Henan University of Science and Technology, Luoyang, People's Republic of China 471003 and Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Scien ...;Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People's Republic of China 100190 and State Key Laboratory of Informa ...
Venue:
Designs, Codes and Cryptography
Year:
2011

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Abstract

In this paper, a combinatorial conjecture about binary strings is proposed. Under the assumption that the proposed conjecture is correct, two classes of Boolean functions with optimal algebraic immunity can be obtained. The functions in first class are bent, and then it can be concluded that the algebraic immunity of bent functions can take all possible values except one. The functions in the second class are balanced, and they have optimal algebraic degree and the best nonlinearity up to now.